Some exact solutions of the oscillatory motion of a generalized second grade fluid in an annular region of two cylinders

The velocity field and the associated shear stress corresponding to the longitudinal oscillatory flow of a generalized second grade fluid,between two infinite coaxial circular cylinders,are determined by means of the Laplace and Hankel transforms.Initially,the fluid and cylinders are at rest and at t = 0+ both cylinders suddenly begin to oscillate along their common axis with simple harmonic motions having angular frequencies 1 and 2.The solutions that have been obtained are presented under integral and series forms in terms of the generalized G and R functions and satisfy the governing differential equation and all imposed initial and boundary conditions.The respective solutions for the motion between the cylinders,when one of them is at rest,can be obtained from our general solutions.Furthermore,the corresponding solutions for the similar flow of ordinary second grade fluid and Newtonian fluid are also obtained as limiting cases of our general solutions.At the end,the effect of different parameters on the flow of ordinary second grade and generalized second grade fluid are investigated graphically by plotting velocity profiles.